Multiple choice

Ten different letters of the English alphabet are given. Out of these a word is formed using $5$ letters with replacement. The probability that atleast one letter is repeated in the word is

  1. $\displaystyle \frac{1}{10^{5}}$
  2. $1-\displaystyle \frac{^{10}P_{5}}{10^{5}}$
  3. $1-\displaystyle \frac{^{10}{P_{5}}}{10!}$
  4. $\displaystyle \frac{^{10}{P_{5}}}{10^{5}}$
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B Correct answer
Explanation

Total possible words = 10^5. Words with no repetition = 10 * 9 * 8 * 7 * 6 = 10P5. Probability of no repetition = 10P5 / 10^5. Probability of at least one repetition = 1 - (10P5 / 10^5).

AI explanation

The total number of possible 5-letter words formed from 10 letters with replacement is 10^5. To find the probability of at least one repeated letter, calculate the complement event where all 5 letters are distinct. The number of ways to form a word with 5 distinct letters from 10 is given by the permutation 10P5. The probability of getting all distinct letters is 10P5 / 10^5. Therefore, the probability of at least one letter being repeated is 1 minus 10P5 / 10^5.

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